Two-scale convergence of Stekloff eigenvalue problems in perforated domains
The author deals with the spectral asymptotics of Stekloff eigenvalue problems for an elliptic linear differential operator \[ \begin{aligned} -\sum_{i,j=1}^{N}\frac{\partial}{\partial x_i}\left(a_{ij}\left(x,\frac{x}{\varepsilon}\right)\frac{\partial u_{\varepsilon}}{\partial x_{j}}\right)=0, &\quad \text{in } \Omega^{\varepsilon},\\ \sum_{i,j=1}^{N} a_{ij}\left(x,\frac{x}{\varepsilon}\right)\frac{\partial u_{\varepsilon}}{\partial x_{j}}\nu_i=\lambda_{\varepsilon}u_{\varepsilon}, &\quad \text{on } \partial T^{\varepsilon},\\ u_{\varepsilon}=0, &\quad \text{on } \partial \Omega,\\ \varepsilon\int_{S^{\varepsilon}}|u_{\varepsilon}|^2\,d\sigma_{\varepsilon}(x)=1, &{}\end{aligned} \] where \(\Omega\) is a bounded open set and \(\Omega^{\varepsilon}\) a perforated domain. By using the two-scale convergence method, a very accurate, precise, and concise homogenization result is obtained. Also, a convergence result for gradients of eigenfunctions is provided, which improves the understanding of the asymptotic behavior of eigenfunctions. Finally, it is justified that the natural local problem is not an eigenvalue problem.
scientific article; zbMATH DE number 5847805